Yes and yes. I'm going to see how much mail I get. Unrelatedly, both my personal and work cell phones have been getting robocalls despite me being on the do not call.
When you declared, the poll worker marked you down as a Democrat in the book, and you signed to vote: that is the paperwork.
And it'll depend what year on if you'll get mail. They'll use microdata analysis and if you are never likely to vote in another primary, they'll probably not waste the time on sending mail. This is particularly true for odd years when there is no elections (except at the state level, which means they'd have to think you are going to be compelled to come out again). In all honesty, primaries look at churning out those voters they know will come out versus using fewer resources on questionable voters.